## Abstract

In the 2-d setting, given an H^{1} solution v(t) to the linear Schrödinger equation i∂_{t}v + v = 0, we prove the existence (but not uniqueness) of an H^{1} solution u(t) to the defocusing nonlinear Schrödinger (NLS) equation i∂_{t} u + u -|u| ^{p-1}u = 0 for nonlinear powers 2 < p < 3 and the existence of an H^{1} solution u(t) to the defocusing Hartree equation i∂_{t} u + u -(|x|^{-γ}black star|u|^{2})u = 0 for interaction powers 1 < γ < 2, such that ||u(t) - v(t)|| _{H}^{1} → 0 as t → + ∞. This is a partial result toward the existence of well-defined continuous wave operators H ^{1} → H^{1} for these equations. For NLS in 2-d, such wave operators are known to exist for p < 3, while for p ≤ 2 it is known that they cannot exist. The Hartree equation in 2-d only makes sense for 0 < γ < 2, and it was previously known that wave operators cannot exist for 0 < γ ≤ 1, while no result was previously known in the range 1 < γ < 2. Our proof in the case of NLS applies a new estimate of CollianderGrillakisTzirakis to a strategy devised by Nakanishi. For the Hartree equation, we prove a new correlation estimate following the method of CollianderGrillakisTzirakis.

Original language | English (US) |
---|---|

Pages (from-to) | 117-138 |

Number of pages | 22 |

Journal | Journal of Hyperbolic Differential Equations |

Volume | 7 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2010 |

## Keywords

- Hartree equation
- Schrödinger equation
- Wave operators

## ASJC Scopus subject areas

- Analysis
- Mathematics(all)

## Fingerprint

Dive into the research topics of 'Asymptotically linear solutions in h^{1}of the 2-d defocusing nonlinear Schrödinger and Hartree equations'. Together they form a unique fingerprint.